Application of Tri-layered RNN scheme for Maxwell model subject to MHD

Sana Ullah Saqib; Yin-Tzer Shih; Muhammad Jahanzaib; Abdul Wahab; Adnan; Shih-Hau Fang; Sana Ullah Saqib; Yin-Tzer Shih; Muhammad Jahanzaib; Abdul Wahab; Adnan; Shih-Hau Fang

doi:10.3934/math.2026038

AIMS Mathematics

2026, Volume 11, Issue 1: 881-906. doi: 10.3934/math.2026038

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Research article

Application of Tri-layered RNN scheme for Maxwell model subject to MHD

1.
Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan
2.
Department of Mathematics, King Fahd University of Petroleum and Minerals (KFUPM), Dhahran 31261, Saudi Arabia
3.
Department of Mathematics, Mohi-ud-Din Islamic University, Nerian Sharif, AJ & K 12080, Pakistan
4.
Department of Electrical Engineering, National Taiwan Normal University, Taipei 106308, Taiwan
5.
Yuan Ze University, Taoyuan 320, Taiwan

Received: 27 October 2025 Revised: 04 December 2025 Accepted: 12 December 2025 Published: 12 January 2026
MSC : 35A22, 76A02, 76A05, 76A20, 76D05, 76S05, 76W05

This study examines the characteristics of the Maxwell model under the influence of magnetohydrodynamics (MHD). Due to its inherent viscosity and elasticity, this model has significant applications in both industrial and biological contexts. The core innovation of this work lies in the development and application of a soft computational approach, specifically, the design and implementation of layered supervised recurrent neural networks optimized via the Levenberg-Marquardt (LSRNNs-LMO) technique, to predict the thermodynamic properties of the Maxwell model over a sheet, with particular focus on melting heat and zero mass flux boundary conditions, as inspired by the Cattaneo-Christov heat flux formulation and Lorentz force effects (TMS-MHZCL) model. The LSRNNs-LMO model is trained using data generated through a reliable numerical scheme. Simulation outcomes from the proposed LSRNNs-LMO method show excellent agreement with numerical results across multiple test cases, exhibiting minimal errors and high robustness. The accuracy of the proposed technique is thoroughly validated using error histograms, optimization control curves (mean squared error), root-mean-square error, autocorrelation analysis, regression evaluation, and the Nash-Sutcliffe efficiency (NSE) metric for the TMS-MHZCL model. These assessments provide strong evidence of the predictive validity and precision of the developed LSRNNs-LMO approach.
- thermal properties of the Maxwell model,
- stretched sheet,
- Lobatto-IIIA scheme,
- Lorentz forces,
- magneto-hydrodynamics,
- recurrent neural network
Citation: Sana Ullah Saqib, Yin-Tzer Shih, Muhammad Jahanzaib, Abdul Wahab, Adnan, Shih-Hau Fang. Application of Tri-layered RNN scheme for Maxwell model subject to MHD[J]. AIMS Mathematics, 2026, 11(1): 881-906. doi: 10.3934/math.2026038

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Abstract

This study examines the characteristics of the Maxwell model under the influence of magnetohydrodynamics (MHD). Due to its inherent viscosity and elasticity, this model has significant applications in both industrial and biological contexts. The core innovation of this work lies in the development and application of a soft computational approach, specifically, the design and implementation of layered supervised recurrent neural networks optimized via the Levenberg-Marquardt (LSRNNs-LMO) technique, to predict the thermodynamic properties of the Maxwell model over a sheet, with particular focus on melting heat and zero mass flux boundary conditions, as inspired by the Cattaneo-Christov heat flux formulation and Lorentz force effects (TMS-MHZCL) model. The LSRNNs-LMO model is trained using data generated through a reliable numerical scheme. Simulation outcomes from the proposed LSRNNs-LMO method show excellent agreement with numerical results across multiple test cases, exhibiting minimal errors and high robustness. The accuracy of the proposed technique is thoroughly validated using error histograms, optimization control curves (mean squared error), root-mean-square error, autocorrelation analysis, regression evaluation, and the Nash-Sutcliffe efficiency (NSE) metric for the TMS-MHZCL model. These assessments provide strong evidence of the predictive validity and precision of the developed LSRNNs-LMO approach.

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